In this paper we consider the classical problem of pursuit and evasion for continuous-time and discrete-time systems. We prove the convergence, as the time step goes to 0, of the upper and lower value functions of the discrete-time game to the upper and lower values of the differential game. This is done assuming a capturability condition either on the differential game, or on the discrete-time game uniformly for small values of the time step. An application is the existence of the value in the sense of Fleming under rather general conditions.